Apparatus for obtaining a conoscopic holograph using incoherent light is described in patent document U.S. Pat. No. 4,602,844. The apparatus described in that document includes, as illustrated diagrammatically in accompanying FIG. 1, a birefrigent uniaxial crystal inserted between two circular polarizers, and a photosensitive element consituting a recording medium.
In Document U.S. Pat. No. 4 602 844, the axis of the crystal is parallel to the geometrical axis of the system, i.e. perpendicular to the recording medium.
The crystal decomposes an incident ray firstly into an ordinary ray subjected to a refractive index n.sub.o, and secondly into an extraordinary ray subjected to a refractive index which varies as a function of the angle of incidence .theta., with this variable refractive index being written n.sub.e (.theta.).
These two rays propagate at different speeds within the crystal. As a result they are at different phases on leaving the crystal. Conoscopic holography is based on the fact that this phase difference is a function of the angle of incidence .theta.. The two rays interfere on the recording medium (photographic film, CCD, . . .) after passing through the outlet polarizer such that the intensity of the resulting ray is also a function of the angle .theta.. In other words, unlike conventional holography, each incident ray produces its own reference ray. The set of rays situated on a cone whose axis is parallel to the optical axis of the crystal and having an aperture angle .theta. will give the same intensity on the observation plane.
As shown in accompanying FIG. 2, the conoscopic hologram of a point obtained by means of the above-mentioned apparatus corresponds to a zoned grating whose transmittance varies sinusoidally as a function of the square of its distance from the center of the grating, i.e. a series of concentric angular interference fringes.
The conoscopic hologram of an object is the superposition of the holograms of each of the points constituting the object. FIGS. 3b and 3c of above-mentioned Document U.S. Pat. No. 4,602,844 respectively show the holograms for two points and for three points of a plane object.
The resulting hologram contains all of the useful information, such that it is possible to reconstruct the initial object in three dimensions.
The conoscopic system performs a linear transformation between the object and its hologram.
The intensity at a point Q of the elementary hologram of the point P is given by: EQU I.sub.p (Q)=I(P)(1+cos.alpha.(P)r.sup.2) (1)
where .alpha.(P), known as the Fresnel parameter, depends on the optico-geometric characteristics of the crystal, on the wavelength .lambda. of the light, and on the longitudinal distance z(P) at which the point P is located relative to the recording plane.
The impulse response of the system, which characterizes the linear transformation, is written: EQU T(x',Y')=1+cos(.alpha.r.sup.2) (2)
where r.sup.2 =x'.sup.2 +y'.sup.2,
and: EQU T.alpha.=2.pi.L.delta.n/.lambda.n.sub.o.sup.2 Z.sub.c.sup.2, (3)
with
.lambda.=source wavelength PA0 L=crystal wavelength along the optical axis PA0 n.sub.o =the ordinary index of the crystal PA0 .delta.n=the absolute value of the difference between the ordinary PA0 x,y,z=coordinates in the object volume PA0 x',y'=coordinates in the hologram plane PA0 Zc=the corrected longitudinal coordinate of P and is given by: EQU Z.sub.c =Z(x,y)-L+L/n.sub.o
index and the extraordinary index
(4)
where Z(x,y) is the distance between the holographic plane and the object point under consideration, situated at the lateral position (x,y). The Fresnel parameter .alpha. can also be written: EQU .alpha.=.pi./.lambda.eq(Z.sub.c)Z.sub.c ( 5)
thus defining an equivalent wavelength .lambda.eq: EQU .lambda.eq=.lambda.n.sub.o.sup.2 Z.sub.c/.delta. n2L (6)
or: EQU .alpha.=.pi./.lambda.f.sub.c ( 7)
thus defining the focal length f.sub.c of the Fresnel lens: EQU f.sub.c= n.sub.0.sup.2 Z.sub.c.sup.2 /.delta.n 2L (8)
When the object under consideration is plane (.alpha.=constant) the equivalent wavelength and the focal length f.sub.c are constants of the system.
Equation (5) then shows that the conoscopic hologram of a point recorded at a wavelength .lambda. is similar to the hologram of the same point recorded using coherent light (Gabor holography) at the equivalent wavelength .lambda.eq. It should be observed that the conoscopic hologram measures intensities and not amplitudes.
Since the distances Z.sub.c and L are of the same order of magnitude, and since .delta.n is about 0.1, the wavelength .lambda.eq is greater than the real wavelength .lambda. at which recording takes place: typically .lambda.eq=3 .mu.m to 100 .mu.m.
As a result, the lateral resolution of the hologram (proportional to the wavelength .lambda.) is less in conoscopic holography than in conventional holography. Its value lies around a few tens of micrometers.
As mentioned above, a hologram recorded using a conoscopic apparatus contains all of the useful information.
For example, for a hologram of a point corresponding to a zoned grating:
the center of the zone and the object point lie on the same straight line parallel to the optical axis, and if the object point is translated transversely or laterally, then the hologram is translated identically in the holographic plane. The coordinates of the center C(x.sub.o, y.sub.o) of the Fresnel zone are thus equal to the first two coordinates of the holographed point P(x.sub.o, y.sub.o, z.sub.o);
the intensity of the hologram gives the light energy in the light aperture cone; and
the spacing of the fringes gives the distance between the object and the observation plane, independently of the position of the conoscopic apparatus.
The following may be written: EQU Z.sub.c =R.sup.2 /F.lambda.eq (9)
and EQU Z(x,y)=Z.sub.c +L-L/n.sub.o =R.sup.2 /F.lambda.eq+L-L/n.sub.o( 10)
where R is the radius of the Fresnel zone and F is the number of light and dark fringes on the radius.
The advantages of the holographic method described above include the following:
there are no additional constraints due to space coherence of the light used for emitting the scene by a conventional standard optical system;
it is inherently stable (each elementary hologram follows the point with which it is associated), thus reducing hologram-taking conditions to those of ordinary photography and making it possible to operate in an industrial environment for moving objects; and
the resolution required for recording is adaptable to CCD sensors making real-time digitizing possible and also enabling a plurality of images to be summed in order to improve the signal to noise ratio.
Since recording is linear, for an object having light-diffusing points lying on a three-dimensional surface S, the intensity at a point Q of the hologram will be the incoherent superposition of all of the elementary holograms of the points P constituting the object and can be written: EQU H(Q)=.intg..sub.S I.sub.p (Q)dP (11)
In the two-dimensional case, this integral reduces to a single convolution, and in the three-dimensional case to a series of convolutions.
Such holograms may be reconstituted either optically by encoding the digital information on a photolithographic plate and reading it back visually by means of a laser beam, or else numerically by applying appropriate deconvolution algorithms for obtaining the file z(x,y).
However, in spite of the great hopes based on conoscopic holography as described above, it has not yet led to industrial developments.
This appears to be due to the fact that it is relatively difficult to make use of a hologram made in this way.
In fact, a conoscopic hologram obtained using the means described in Document U.S. Pat. No. 4,602,844 contains two types of interfering information corresponding respectively to a coherent continuous background or "bias" representing non-diffracted light, and to a conjugate image, both of which degrade the basic information which is sufficient on its own for reconstructing the object.
These two types of interfering information which are superposed on the useful information when a conoscopic hologram is recorded can be shown up by illuminating the conoscopic hologram recorded on a photosensitive film by means of a monochromatic plane wave. Three diffractive beams are then observed: the first beam represents the wave transmitted directly through the film and corresponds to the bias; the second wave is a spherical wave diverging from a virtual object which is a replica of the original object; and the third wave is a spherical wave converging on a conjugate real image of the object situated symmetrically to the virtual image about the plane of the hologram.
The two above-mentioned interfering types of information (bias and conjugate image) can also be shown up by the following, more theoretical approach.
For plane objects, the linear transformation between the intensity I of x, y of the object and the intensity H(x', y') of the hologram is given by the convolution: EQU H(x',y')=I(x,y)* T(x,y) (12)
After developing the convolution equation (12), the hologram appears at a Fresnel transform: EQU H(x',y')=I.sub.o +I(x,y)* cos(.alpha.r.sup.2) (13)
or EQU H(x',y')=I.sub.o +1/2I(x,y)* e.sup.j.alpha.r.spsp.2 +1/2I(x,y)* e.sup.-j.alpha.r.spsp.2 ( 14)
where I.sub.o represents the bias intensity which penetrates directly through the system and where 1/2I(x,y) * e.sup.-j.alpha.r.spsp.2 represents the conjugate image.
As described in the document optics Communications Vol. 65, No. 4, Feb. 15, 1988, pp. 243-249, proposals have been made to overcome the bias and the conjugate image by inclining the optical axis of the birefringent crystal relative to the optical axis of the system. However, as described in that document, this configuration is incapable of restoring the object in full and it is accompanied by all or a portion of the spectrum being deteriorated. Since this configuration gives very mediocre results, it was quickly abandoned.
Another solution for eliminating the conjugate image and the bias is described in French patent application number 88 17225 filed Dec. 27, 1988. It is based on the linear combination of several holograms.
In the basic disposition described in above-outlined Document U.S. Pat. No. 4,602,844, the inlet and outlet circular polarizers are constituted by a rectilinear polarizer and a quarterwave plate which are integrally fixed together. The invention described in patent application FR-88 17225 proposes separating these two plates, thereby making it possible to impart any polarization to the transmitted wave. Accompanying FIG. 3 is a diagram of the resulting conoscopic element. The angles .phi..sub.0, .phi..sub.1, .phi..sub.3 represent the positions of the main polarization axes of the various plates. If the optical impulse response is written T, the intensity at a point Q of the hologram of a point P is written: EQU I.sub.p (Q)=I(P)T(P,Q) (15 )
The table of accompanying FIG. 4 shows the various expressions for the function T depending on the values of .alpha..sub.1 =.phi..sub.0 -.phi..sub.1 and of .alpha..sub.2 =.phi..sub.2 -.phi..sub.3. The angle .psi. and the distance r represent the polar coordinates of the point Q in a frame of reference based on the center of the zoned grid (orthogonal projection of P on the recording plane). The parameter .alpha.(P) is defined in the same way as in the basic configuration.
It can be seen that it is possible to use simple linear combinations of the transfer functions of the above table to obtain the following impulse responses: EQU T.sub.c (P,Q)=cos.alpha.(P)r.sup.2 ( 16) EQU T.phi..sub.o (P,Q)=sin2(.psi.-.phi..sub.o) sin.alpha.(P)r.sup.2( 17)
of, using .phi..sub.o =0 and .phi..sub.o 32 .pi./4, respectively EQU T.sub.c (P,Q)=cos.alpha.(P)r.sup.2 ( 18) EQU T.sub.o (P,Q)=sin2.psi.sin.alpha.(P)r.sup.2 ( 19) EQU T.pi./4(P,Q)=cos2.psi.sin.alpha.(P)r.sup.2 ( 20)
By using linear combinations of these three functions in the Fourier plane, it is possible to obtain the complex impulse response: EQU T(P,Q)=expj.alpha.(P)r.sup.2 ( 21)
exactly for a point or a plane object.
However, for a three-dimensional surface, this expression is no longer applicable and it can be considered as being valid only for the high frequencies of the hologam. Methods exist enabling the low frequencies to be restored, in particular by using a two-dimensional image of the scene as provided by another camera. Such methods are complex to implement and require the knowledge of a normalization factor between the image and the hologram, and this factor is difficult to determine.
Consequently, this type of holography is of practical use only for applications where low frequencies are not of interest, in particular system for identifying points (projecting an array of points) or for telemetry.
The object of the present invention is to provide new means for obtaining a complex hologram for an arbitrary three-dimensional surface merely by filtering in the Fourier plane.
As shown below, the present invention serves to eliminate the conjugate image without any correction depending on its longitudinal coordinate.